This function computes the n-dimensional discrete Fourier Transform
over any axes in an M-dimensional array by means of the
Fast Fourier Transform (FFT). By default, the transform is computed over
the last two axes of the input array, i.e., a 2-dimensional FFT.

Parameters :

a : array_like

Input array, can be complex

s : sequence of ints, optional

Shape (length of each transformed axis) of the output
(s[0] refers to axis 0, s[1] to axis 1, etc.).
This corresponds to n for fft(x, n).
Along each axis, if the given shape is smaller than that of the input,
the input is cropped. If it is larger, the input is padded with zeros.
if s is not given, the shape of the input (along the axes specified
by axes) is used.

axes : sequence of ints, optional

Axes over which to compute the FFT. If not given, the last two
axes are used. A repeated index in axes means the transform over
that axis is performed multiple times. A one-element sequence means
that a one-dimensional FFT is performed.

Returns :

out : complex ndarray

The truncated or zero-padded input, transformed along the axes
indicated by axes, or the last two axes if axes is not given.

Shifts zero-frequency terms to the center of the array. For two-dimensional input, swaps first and third quadrants, and second and fourth quadrants.

Notes

fft2 is just fftn with a different default for axes.

The output, analogously to fft, contains the term for zero frequency in
the low-order corner of the transformed axes, the positive frequency terms
in the first half of these axes, the term for the Nyquist frequency in the
middle of the axes and the negative frequency terms in the second half of
the axes, in order of decreasingly negative frequency.

See fftn for details and a plotting example, and numpy.fft for
definitions and conventions used.